What is 8.127142714271427142... as a Fraction?

8.127142 equals 81263399990 as a fraction.

How do you turn 8.127142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 8.127142 repeating into a fraction, begin writing this simple equation:

n = 8.127142 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (7142) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 8127142.7142 (equation 2)

Step 3: Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out.

1000000 × n = 8127142.7142
    100 × n = 8.127142
 999900 × n = 8126330

8126330999900 could be the answer, but it still can be put in the simplest form, i.e., reduced.

To simplify this fraction, divide the numerator and denominator by 10 (the GCF - greatest common factor).

n = 8126330999900 = 8126330 ÷ 10999900 ÷ 10 = 81263399990. So,

8.127142 = 81263399990 as the lowest possible fraction.

As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus 81263399990 is also equal to 81271399990 when expressed as a mixed number.

The repeating decimal 8.127142 (vinculum notation) has a repeated block length of 4. It is also represented as 8.12714271427142... (ellipsis notation) which equals approximately 8.1271427142714271427142 (decimal approximation)(*).

The recurring decimal 8.127142 can be written as a ratio of two integers having 812633 as the numerator and 99990 as the denominator. So, it is a rational number (named after ratio). It can be shown that a number is rational if its decimal representation is repeating or terminating.

(*) At present, there is no single universally accepted notation or phrasing for repeating decimals.